License	MIT
Maintainer	douglas.mcclean@gmail.com
Stability	experimental
Safe Haskell	Safe
Language	Haskell2010
Extensions	ParallelListComp RankNTypes ExplicitForAll

Data.ExactPi

Contents

Utils

Description

This type is sufficient to exactly express the closure of Q ∪ {π} under multiplication and division. As a result it is useful for representing conversion factors between physical units. Approximate values are included both to close the remainder of the arithmetic operations in the Num typeclass and to encode conversion factors defined experimentally.

Synopsis

data ExactPi
- = Exact Integer Rational
- | Approximate (forall a. Floating a => a)
approximateValue :: Floating a => ExactPi -> a
isZero :: ExactPi -> Bool
isExact :: ExactPi -> Bool
isExactZero :: ExactPi -> Bool
isExactOne :: ExactPi -> Bool
areExactlyEqual :: ExactPi -> ExactPi -> Bool
isExactInteger :: ExactPi -> Bool
toExactInteger :: ExactPi -> Maybe Integer
isExactRational :: ExactPi -> Bool
toExactRational :: ExactPi -> Maybe Rational
rationalApproximations :: ExactPi -> [Rational]
getRationalLimit :: Fractional a => (a -> a -> Bool) -> [Rational] -> a

Documentation

data ExactPi Source #

Represents an exact or approximate real value. The exactly representable values are rational multiples of an integer power of pi.

Constructors

Exact Integer Rational	`Exact z q` = q * pi^z. Note that this means there are many representations of zero.
Approximate (forall a. Floating a => a)	An approximate value. This representation was chosen because it allows conversion to floating types using their native definition of `pi`.

Instances

Floating ExactPi Source #
Instance details Defined in Data.ExactPi Methods pi :: ExactPi # exp :: ExactPi -> ExactPi # log :: ExactPi -> ExactPi # sqrt :: ExactPi -> ExactPi # (**) :: ExactPi -> ExactPi -> ExactPi # logBase :: ExactPi -> ExactPi -> ExactPi # sin :: ExactPi -> ExactPi # cos :: ExactPi -> ExactPi # tan :: ExactPi -> ExactPi # asin :: ExactPi -> ExactPi # acos :: ExactPi -> ExactPi # atan :: ExactPi -> ExactPi # sinh :: ExactPi -> ExactPi # cosh :: ExactPi -> ExactPi # tanh :: ExactPi -> ExactPi # asinh :: ExactPi -> ExactPi # acosh :: ExactPi -> ExactPi # atanh :: ExactPi -> ExactPi # log1p :: ExactPi -> ExactPi # expm1 :: ExactPi -> ExactPi # log1pexp :: ExactPi -> ExactPi # log1mexp :: ExactPi -> ExactPi #
Fractional ExactPi Source #
Instance details Defined in Data.ExactPi Methods (/) :: ExactPi -> ExactPi -> ExactPi # recip :: ExactPi -> ExactPi # fromRational :: Rational -> ExactPi #
Num ExactPi Source #
Instance details Defined in Data.ExactPi Methods (+) :: ExactPi -> ExactPi -> ExactPi # (-) :: ExactPi -> ExactPi -> ExactPi # (*) :: ExactPi -> ExactPi -> ExactPi # negate :: ExactPi -> ExactPi # abs :: ExactPi -> ExactPi # signum :: ExactPi -> ExactPi # fromInteger :: Integer -> ExactPi #
Show ExactPi Source #
Instance details Defined in Data.ExactPi Methods showsPrec :: Int -> ExactPi -> ShowS # show :: ExactPi -> String # showList :: [ExactPi] -> ShowS #
Semigroup ExactPi Source #	The multiplicative semigroup over `Rational`s augmented with multiples of `pi`.
Instance details Defined in Data.ExactPi Methods (<>) :: ExactPi -> ExactPi -> ExactPi # sconcat :: NonEmpty ExactPi -> ExactPi # stimes :: Integral b => b -> ExactPi -> ExactPi #
Monoid ExactPi Source #	The multiplicative monoid over `Rational`s augmented with multiples of `pi`.
Instance details Defined in Data.ExactPi Methods mempty :: ExactPi # mappend :: ExactPi -> ExactPi -> ExactPi # mconcat :: [ExactPi] -> ExactPi #

approximateValue :: Floating a => ExactPi -> a Source #

Approximates an exact or approximate value, converting it to a Floating type. This uses the value of pi supplied by the destination type, to provide the appropriate precision.

isZero :: ExactPi -> Bool Source #

Identifies whether an ExactPi is an exact or approximate representation of zero.

isExact :: ExactPi -> Bool Source #

Identifies whether an ExactPi is an exact value.

isExactZero :: ExactPi -> Bool Source #

Identifies whether an ExactPi is an exact representation of zero.

isExactOne :: ExactPi -> Bool Source #

Identifies whether an ExactPi is an exact representation of one.

areExactlyEqual :: ExactPi -> ExactPi -> Bool Source #

Identifies whether two ExactPi values are exactly equal.

isExactInteger :: ExactPi -> Bool Source #

Identifies whether an ExactPi is an exact representation of an integer.

toExactInteger :: ExactPi -> Maybe Integer Source #

Converts an ExactPi to an exact Integer or Nothing.

isExactRational :: ExactPi -> Bool Source #

Identifies whether an ExactPi is an exact representation of a rational.

toExactRational :: ExactPi -> Maybe Rational Source #

Converts an ExactPi to an exact Rational or Nothing.

rationalApproximations :: ExactPi -> [Rational] Source #

Converts an ExactPi to a list of increasingly accurate rational approximations. Note that Approximate values are converted using the Real instance for Double into a singleton list. Note that exact rationals are also converted into a singleton list.

Implementation is based on Chudnovsky's algorithm.

Utils

getRationalLimit :: Fractional a => (a -> a -> Bool) -> [Rational] -> a Source #

Given an infinite converging sequence of rationals, find their limit. Takes a comparison function to determine when convergence is close enough.

>>> getRationalLimit (==) (rationalApproximations (Exact 1 1)) :: Double
3.141592653589793